Properties

Label 207.3.j.a
Level $207$
Weight $3$
Character orbit 207.j
Analytic conductor $5.640$
Analytic rank $0$
Dimension $30$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [207,3,Mod(10,207)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(207, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("207.10");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 207 = 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 207.j (of order \(22\), degree \(10\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.64034147226\)
Analytic rank: \(0\)
Dimension: \(30\)
Relative dimension: \(3\) over \(\Q(\zeta_{22})\)
Twist minimal: no (minimal twist has level 23)
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 30 q + 11 q^{2} - 23 q^{4} + 11 q^{5} - 11 q^{7} - 10 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q + 11 q^{2} - 23 q^{4} + 11 q^{5} - 11 q^{7} - 10 q^{8} - 11 q^{10} + 11 q^{11} - 11 q^{13} + 11 q^{14} + 73 q^{16} - 44 q^{17} + 22 q^{19} - 77 q^{20} - 36 q^{23} - 152 q^{25} + 186 q^{26} - 275 q^{28} + 88 q^{29} - 110 q^{31} + 147 q^{32} + 231 q^{34} - 209 q^{35} + 341 q^{37} - 374 q^{38} + 429 q^{40} - 77 q^{41} + 77 q^{43} - 110 q^{44} - 99 q^{46} + 110 q^{47} - 422 q^{49} + 396 q^{50} - 472 q^{52} + 187 q^{53} - 165 q^{55} - 176 q^{56} - 13 q^{58} + q^{59} + 297 q^{61} - 82 q^{62} + 386 q^{64} - 220 q^{65} + 11 q^{67} - 198 q^{70} + 176 q^{71} - 121 q^{73} + 352 q^{74} + 110 q^{76} - 110 q^{77} + 33 q^{79} + 242 q^{80} + 96 q^{82} + 154 q^{83} + 275 q^{85} - 143 q^{86} + 429 q^{88} - 121 q^{89} - 166 q^{92} - 295 q^{94} + 154 q^{95} + 154 q^{97} - 77 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1 −1.59301 1.83844i 0 −0.272894 + 1.89802i 2.69128 + 1.22907i 0 −1.33225 + 4.53722i −4.26162 + 2.73877i 0 −2.02769 6.90566i
10.2 0.881085 + 1.01683i 0 0.311634 2.16747i 2.08252 + 0.951056i 0 −2.90589 + 9.89656i 7.00598 4.50247i 0 0.867820 + 2.95552i
10.3 1.38322 + 1.59632i 0 −0.0656849 + 0.456848i −6.26010 2.85889i 0 2.54176 8.65643i 6.28757 4.04078i 0 −4.09539 13.9476i
19.1 −0.952843 2.08644i 0 −0.825860 + 0.953093i −3.01510 + 4.69158i 0 −2.67922 + 0.385214i −6.02773 1.76990i 0 12.6616 + 1.82046i
19.2 0.282292 + 0.618133i 0 2.31704 2.67401i 3.24760 5.05336i 0 −3.20136 + 0.460286i 4.91504 + 1.44319i 0 4.04042 + 0.580925i
19.3 1.55977 + 3.41542i 0 −6.61275 + 7.63152i −2.90325 + 4.51755i 0 7.13192 1.02541i −21.9686 6.45058i 0 −19.9577 2.86949i
28.1 −1.44626 0.424661i 0 −1.45368 0.934223i 1.77862 + 0.255727i 0 2.68734 1.22727i 5.65401 + 6.52507i 0 −2.46375 1.12516i
28.2 1.80062 + 0.528710i 0 −0.402313 0.258551i −5.05070 0.726181i 0 −8.85488 + 4.04389i −5.50346 6.35133i 0 −8.71046 3.97793i
28.3 3.10125 + 0.910608i 0 5.42352 + 3.48548i 5.40865 + 0.777647i 0 −0.889564 + 0.406250i 5.17926 + 5.97719i 0 16.0654 + 7.33684i
37.1 −1.44626 + 0.424661i 0 −1.45368 + 0.934223i 1.77862 0.255727i 0 2.68734 + 1.22727i 5.65401 6.52507i 0 −2.46375 + 1.12516i
37.2 1.80062 0.528710i 0 −0.402313 + 0.258551i −5.05070 + 0.726181i 0 −8.85488 4.04389i −5.50346 + 6.35133i 0 −8.71046 + 3.97793i
37.3 3.10125 0.910608i 0 5.42352 3.48548i 5.40865 0.777647i 0 −0.889564 0.406250i 5.17926 5.97719i 0 16.0654 7.33684i
109.1 −0.952843 + 2.08644i 0 −0.825860 0.953093i −3.01510 4.69158i 0 −2.67922 0.385214i −6.02773 + 1.76990i 0 12.6616 1.82046i
109.2 0.282292 0.618133i 0 2.31704 + 2.67401i 3.24760 + 5.05336i 0 −3.20136 0.460286i 4.91504 1.44319i 0 4.04042 0.580925i
109.3 1.55977 3.41542i 0 −6.61275 7.63152i −2.90325 4.51755i 0 7.13192 + 1.02541i −21.9686 + 6.45058i 0 −19.9577 + 2.86949i
136.1 −1.20995 0.777587i 0 −0.802326 1.75685i 2.65558 9.04406i 0 −5.31379 4.60443i −1.21408 + 8.44409i 0 −10.2457 + 8.87791i
136.2 −0.0163142 0.0104845i 0 −1.66150 3.63819i −1.45779 + 4.96477i 0 −1.27914 1.10838i −0.0220779 + 0.153555i 0 0.0758357 0.0657120i
136.3 1.94274 + 1.24852i 0 0.553766 + 1.21258i −0.682875 + 2.32566i 0 6.72814 + 5.82996i 0.876504 6.09622i 0 −4.23029 + 3.66556i
145.1 −1.59301 + 1.83844i 0 −0.272894 1.89802i 2.69128 1.22907i 0 −1.33225 4.53722i −4.26162 2.73877i 0 −2.02769 + 6.90566i
145.2 0.881085 1.01683i 0 0.311634 + 2.16747i 2.08252 0.951056i 0 −2.90589 9.89656i 7.00598 + 4.50247i 0 0.867820 2.95552i
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.d odd 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 207.3.j.a 30
3.b odd 2 1 23.3.d.a 30
12.b even 2 1 368.3.p.a 30
23.d odd 22 1 inner 207.3.j.a 30
69.g even 22 1 23.3.d.a 30
69.g even 22 1 529.3.b.b 30
69.h odd 22 1 529.3.b.b 30
276.j odd 22 1 368.3.p.a 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.3.d.a 30 3.b odd 2 1
23.3.d.a 30 69.g even 22 1
207.3.j.a 30 1.a even 1 1 trivial
207.3.j.a 30 23.d odd 22 1 inner
368.3.p.a 30 12.b even 2 1
368.3.p.a 30 276.j odd 22 1
529.3.b.b 30 69.g even 22 1
529.3.b.b 30 69.h odd 22 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{30} - 11 T_{2}^{29} + 78 T_{2}^{28} - 411 T_{2}^{27} + 1728 T_{2}^{26} - 6157 T_{2}^{25} + 18695 T_{2}^{24} - 49567 T_{2}^{23} + 123051 T_{2}^{22} - 279029 T_{2}^{21} + 570634 T_{2}^{20} - 1009889 T_{2}^{19} + \cdots + 38809 \) acting on \(S_{3}^{\mathrm{new}}(207, [\chi])\). Copy content Toggle raw display